78 TROPOSPHERIC REFRACTION 



to study the differences between individual observed N(h) profiles and the 

 mean profiles predicted by the exponential model, a variable called the 

 A unit has been developed; it is defined simply as the sum of the observed 

 A'' at any height, h, and the refractivity drop from the surface to the 

 height, h, which is predicted by the exponential profile for a given value 

 of A^.. 



Thus 



A{Ns,h) = N(h) + Ns (l-exp{ -c,/i}). (3.46) 



Thus (3.46) adds to N{h) the average decrease of N with height, so that 

 if a particular profile should happen, by coincidence, to be the same as 

 the corresponding exponential profile, the value of A {N s,h) for this pro- 

 file would be equal to N s for all heights. The above analysis shows that 

 the difference between A{Ns,h) from N s, 8A{Ns,h), is a measure of the 

 departure of N(h) from the normal, exponential profile: 



5.4(A^,,^) = AiNs,h) - Ns = A^(/i)— iV. exp{ -c./i}. (3.47) 



It seems logical that the application of the A unit to bending would indi- 

 cate the departures of bending from normal, in some way, just as it indi- 

 cates departures of refractivity, A'^, from normal. This is indeed the case, 

 as can be seen in figure 3.16, where for an A'^s = 313.0 exponential atmos- 

 phere, .4(313.0, h) is plotted on one set of graphs for various typical air 

 masses, and the corresponding bending departures from normal are shown 

 in the second set of graphs corresponding to the same air masses. Ob- 

 viously, the bending departures between layers are highly analogous to 

 the A unit variation. It can be seen from figure 3.16 that the similarity 

 exists, although it is less, for higher initial elevation angles. The simi- 

 larity also decreases with increasing height, owing to the fact that the 

 bending departures from normal are an integrated effect, and at low initial 

 elevation angles are more sensitive to A^ variations at the lower heights. 

 This causes an apparent damping of the bending departures from normal 

 at greater heights. However, the A-unit variation is not similarly in- 

 fluenced; hence, a loss of similarity arises at large heights above the earth's 

 surface. 



If (3.46) is differentiated and substituted into (3.2), the following equa- 

 tion results: 



k = Uk -T Uk+1 



(rad) [rad) (.rad) (rad) 



^A{N. 



'k+\ 



X 10-', (3.48) 



1 



